What happens if a full team leaves the game
The game keeps going, 0v5.
The opposing team will just have to destroy your nexus to end the game.
You can see this in the following pro match, where one of the teams decided to surrend before the 20 minutes mark, having had a really bad early game :
What happens if all players in the game leave
This one may be a bit trickier to answer, as there is no mean to trace the game once all player left.
Based on the behavior if a full team leaves, I guess the game would still continue, until one of the team has his minions destroying the ennemy nexus (which will happen at one point, try it in a custom game).
As suggested by Lyrion, it seems that if all 10 players are disconnected from a game (or leave the game on purpose), they will not be able to reconnect. This could imply that the game has been ended. However, this observation was made in a custom game, and this behavior may be specific for these games.
Although Veigar is a great burst-nuke damage mid who can counter a bunch of AP champions, with one of the best non-ultimate stuns in the game, and never-ending AP farming with his Q, he does have some flaws that may make him a unappealing pick in competitive play.
His damage comes in a burst which is good for killing a champion, but makes him fairly useless when his skills are on cooldown. He is not like Ryze who can deal consistent damage throughout a whole team fight.
Similar to the first point, his cooldowns are fairly long. His Q being the lowest at 6 seconds, his W at 10 seconds, and his E (The AoE stun) being at 16 seconds when fully leveled.
Veigar can be difficult to learn due to his high skill cap. He has to manage his Q's ability to give him AP for last hitting with it and utilizing it for poke. His E (stun) and his W (dark matter) can be hard to get used to. At level one his stun will stun the opponent for only 1.5 seconds, and his W takes 1.2 seconds to fall. That gives you 0.3 seconds to get your W off after your E before your opponent can dodge it.
Veigar is quite squishy, if the opposing team can initiate on you before you can do your combo you will die very quickly. Veigar has the ability to turn a teamfight into a 4v5, either by bursting down an opponent or by getting bursted himself.
Veiagr also has no ways to escape like other AP assassins. His stun can be hard to aim properly, and can be avoided by walking around it. Also in the early game his stun only last 1.5 seconds, which is not muck time to get away from champions with gap closers.
The best way in my opinion to view this, is that Veigar is an assassin and can either do very well or very poorly. Therefore, it is a gamble to pick him which is why he isn't picked very often in competetive play.
Best Answer
From the LCS rules book, you can read under point 3.4 Substitutions:
To me it sounds like you can swap a player mid game, but only if there is a serious emergency.
I am not quite sure how the game you mentioned continued, but a disconnect can happen due to various reasons like a bluescreen, power-outage and so on.